How to Read a Math Book

Summary:
You can’t read a math book the way you read other books.
It takes a special approach to read math, not just pass your eyes over it.
You may need as long as half an hour to get through one page, but
you will understand it when you’re done.

Two Kinds of Reading

In
a Usenet article,
Chan-Ho Suh neatly summed up
the “two kinds of understanding a math text. The first is in being
able to follow, reconstruct, etc. a proof [or solution method]. The
second lies in rewiring one’s brain so that it’s obvious why the
theorem is true” or why the solution works. “Contrary to impressions
you may have garnered from your math courses, the first kind of
understanding is not the primary goal, ... but the second kind
definitely is.”

What Not to Do

Don’t memorize too much. Many students try to commit
everything to memory. Math doesn’t work like that: you are learning
methods of problem solving. If you treat the material like a mass of
unrelated facts, it will be a big jumble in your head and you’ll very
naturally feel frustrated.

True, there are a few formulas you’ll need to have available. Write
them down on one “cheat sheet”. Look at them in odd moments and
practice saying them over to yourself. Then write
them down
at the start of any test.

Don’t start with the homework problems. The assigned
problems are not “the homework”. They’re the last part of the
homework. The first part is reading and understanding today’s section
of the textbook.

Most students can’t just listen to the lecture and then
immediately solve the homework problems. The textbook is there to
reinforce your understanding of what was covered in class and to
explain points the lecture didn’t have time to cover.

Don’t be afraid to go back. Math is relentlessly
cumulative. If you don’t understand something in today’s lesson,
perhaps it’s because you didn’t really understand something in an
earlier lesson, or you’ve forgotten it. Go back and re-learn that
earlier material.

Granted, it can be discouraging to go back to an earlier section
of the book. But you are not moving backward. You’re still making
progress because you’re strengthening your understanding of that
earlier concept.

What to Do

First skim for an overview. Nobody understands
something complex on first reading: about all you can hope to do is
get a general sense of what it’s about and perhaps one or two of the
main points of the argument. That doesn’t mean there’s anything wrong
with you; it’s how almost everyone’s mind works. That first reading
just lays the framework for you to fill in later with details.

Then reread with concentration. Read slower this time.
Highlight important points for further study. (Don’t overdo
the highlighting. Distinguish key points and highlight them; that
should be well under 50% of the text. Color-code if that helps you.)

Some students find it helpful to take notes on separate paper while
reading. If you’ve never done this, you might want to give it a try.
You may find that it helps focus your mind and fit the concepts
together better.

Go through each step of each example. Don’t read; write.
Many examples have some steps left out; you should write down a
complete solution with all steps. Make sure you understand how the
book gets from each step to the next step.

This is where many students shortchange themselves. There’s a huge
tendency to hit an example and have the eyes just jump to the next bit
of English. But remember that you learn math by
doing, not by reading. You need to work each example to understand
the concept.

Fill in any gaps. If there are still any words or
concepts you don’t understand, go back to the book and learn them. If
you need to, put them on the list to talk over with your study buddy
or with a Baker Center tutor,
or see your instructor outside of class if s/he’s available.

Whatever you do, get those problems cleared up before the next
class so that they don’t interfere with your understanding that
lecture.

Think about what you’ve read. Fit it in with what
you already know. Is this a more general version of a specific case you
learned earlier? Is this a shorter method for something you previously
learned to do a longer way? Does this use something you learned
previously that looked useless at the time?

Make it your own. Can you explain this to someone
else? (This is where a study group
really shines.) If there’s no one else available, can you explain it
aloud, without stumbling? If you can do that, you probably understand
it. Don’t shortchange yourself here! “I sort of get it” means you
need to go over it again (but maybe after a break).

Divide and conquer. If you have a whole chapter
to learn, master one section at a time. Try to spread your work evenly
over the available time.

Do the homework problems. If you don’t understand how to
do one, look back in the book for a similar problem. Don’t just push
numbers at it; make sure you understand the example and see how to
apply it.